Book Description

In the first chapter we demonstrate that, while in most current data modeling for time-dynamic systems, one works witha pre-specified differential equation and attempts to fit its parameters, in the case of functional data, the equation itself can be inferred from the data. Only assuming that the dynamics are described by a first order nonlinear stochastic differential equation, we obtain these data-adaptive dynamic equations via a simple smoothing-based procedure from the observed data. They reveal the basic dynamics of the processes that generate the observed functional data. We prove consistency and introduce diagnostics to ascertain the fraction of variance that is explained by the deterministic part of the equation. This approach is shown to yield useful insights into the time-dynamic nature of human growth. In the second chapter, we propose a nonparametric method to explicitly model and represent the derivatives of smooth underlying trajectories for longitudinal data. This representation is based on a direct Karhunen-Loève expansion of the unobserved derivatives and leads to the notion of derivative principal components, which complement functional principal components, one of the most popular tools of functional data analysis. The proposed derivative principal components can be obtained for irregularly spaced and sparsely observed longitudinal data, as typically encountered in biomedical studies, as well as for noisy longitudinal data measured on regular and dense grids. We compare the proposed representations for derivatives with alternative approaches via simulation studies, and show that representations by derivative principal components can recover the underlying derivatives more accurately in various settings. As an example, we demonstrate the utility of derivative principal components for the classification of wheat spectra, where the spectral measurements are densely and regularly spaced. In this application, derivative principal components are found to be more predictive for the moisture content of wheat than the conventional functional principal components, and to have improved classification performance. Consistency results and asymptotic convergence rates for the proposed estimates of derivative principal components and other components of the model are derived under mild conditions, using tools from functional data analysis. In the last chapter, we consider a functional regression setting and introduce novel representations to improve prediction accuracy. In various functional regression settings one observes i.i.d. samplesof paired stochastic processes (X,Y), and is interested in predicting the trajectory of Y, given the trajectory X. For example, one may wish to predict the future of a process fromobserving an initial segment of the trajectory. Commonly used functional regression models are based on representations that are obtained separately for X and Y such as functional principal components. In contrast to these established methods, we base our approach on a singular expansion of the paired processes X, Y with singular functions that are derived from the cross-covariance surface between X and Y. The motivation for this approach is that the resulting singular components are better tuned towards reflecting the association between X and Y. The regression relationship is then based onthe assumption that each singular component of Y, or each scalar Y(t), follows an additive regression model with the singular components of X as predictors. The resulting singular additive models are fitted by smooth backfitting. We study the behavior of an implementation of this approach in simulations and data analysis.